Question

The ratio of \(\frac{2.7}{1.2}\) is equivalent to the ratio of \(\frac{b}{4.8}\). What is the value of \(b\) ? A) \(2.13\) B) 4 C) \(6.3\) D) \(10.8\) $$ 60=15 m n+20 $$

Short answer

The value of b is \(10.8\). (Answer choice D)

Step-by-step solution

Set the ratios equal to each other

Given the proportion, we have \(\frac{2.7}{1.2} = \frac{b}{4.8}\)

Cross-multiply

To find the value of \(b\), cross-multiply: (2.7)(4.8) = (1.2)(b)

Solve for \(b\)

Now simplify this equation and solve for \(b\): 12.96 = 1.2b Divide both sides by 1.2: \(b = \frac{12.96}{1.2}\)

Calculate the value of \(b\)

Perform the division: \(b = 10.8\)

Choose the correct answer

Based on the calculations, we can now conclude that the correct answer is:\ D) \(10.8\)

Key concepts

Cross-Multiplication

Understanding cross-multiplication is crucial in comparing ratios and solving for unknown variables in proportion problems.

Cross-multiplication is a technique that allows us to find the value of the unknown in a proportion by multiplying diagonally and creating an equation. When given two ratios, such as \( \frac{a}{b} = \frac{c}{d} \), cross multiplication involves multiplying the numerator of one ratio by the denominator of the other and setting the products equal to each other: \( a \times d = b \times c \).

It can be visualized by drawing an 'X', hence the name 'cross'-multiplication. The point of intersection represents the equality of the two products. This technique works because it effectively uses the property of equality of fractions: if two fractions are equal, their cross products are also equal.

Minimizing Errors in Cross-Multiplication

• Always ensure that the ratios are in their simplest form before cross-multiplying.
• Check that you're multiplying across correctly: numerator to denominator.
• Double-check your arithmetic to avoid simple errors.

Solving Proportions

Solving proportions involves finding the missing number in a set of equivalent fractions or ratios. To solve this type of mathematical problem, one must possess the skill to manipulate proportions and apply cross-multiplication effectively.

With the example given in the exercise, \( \frac{2.7}{1.2} = \frac{b}{4.8} \), solving for the unknown \(b\) calls for cross-multiplication. After setting up the cross products, the next step involves simplifying the equation to isolate the unknown. In this case, you divide both sides of the equation by 1.2 to find the value of \(b\).

Steps to Problem-Solving in Proportions:

• Identify the known and unknown quantities in the proportion.
• Set up an equation with cross products.
• Perform the arithmetic needed to isolate the unknown variable.
• Check your solution by plugging it back into the original proportion.

SAT Math Problems

The SAT math section tests a student's ability to solve a variety of mathematical problems, including those involving ratios and proportions, like the given exercise. These problems require a clear understanding of algebraic manipulation and number operations.

For success on these types of SAT questions, practice with cross-multiplication and proportion-solving strategies is vital. Being able to recognize when to use these techniques quickly and knowing the shortcuts, such as cross-multiplication, is essential. Furthermore, the SAT often presents problems with extra information or framed in real-life contexts, so students must learn to sift through the information and extract the relevant ratios or equations.

Key SAT Math Problem-Solving Tips:

• Familiarize yourself with different problem types that involve ratios and proportions.
• Practice under timed conditions to simulate actual test scenarios.
• Review mistakes carefully to understand where you might have gone wrong and avoid similar errors.
• Remember shortcut techniques such as cross-multiplication to save time.